Understanding Analysis

1 The Real Numbers.- 1.1 Discussion: The Irrationality of % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaaSbdaGcaa % qaaiaaikdaaSqabaaaaa!3794! $$/sqrt 2 $$.- 1.2 Some Preliminaries.- 1.3 The Axiom of Completeness.- 1.4 Consequences of Completeness.- 1.5 Cantor's Theorem.- 1.6 Epilogue.- 2 Sequences and Series.- 2.1 Discussion: Rearrangements of Infinite Series.- 2.2 The Limit of a Sequence.- 2.3 The Algebraic and Order Limit Theorems.- 2.4 The Monotone Convergence Theorem and a First Look at Infinite Series.- 2.5 Subsequences and the Bolzano-Weierstrass Theorem.- 2.6 The Cauchy Criterion.- 2.7 Properties of Infinite Series.- 2.8 Double Summations and Products of Infinite Series.- 2.9 Epilogue.- 3 Basic Topology of R.- 3.1 Discussion: The Cantor Set.- 3.2 Open and Closed Sets.- 3.3 Compact Sets.- 3.4 Perfect Sets and Connected Sets.- 3.5 Baire's Theorem.- 3.6 Epilogue.- 4 Functional Limits and Continuity.- 4.1 Discussion: Examples of Dirichlet and Thomae.- 4.2 Functional Limits.- 4.3 Combinations of Continuous Functions.- 4.4 Continuous Functions on Compact Sets.- 4.5 The Intermediate Value Theorem.- 4.6 Sets of Discontinuity.- 4.7 Epilogue.- 5 The Derivative.- 5.1 Discussion: Are Derivatives Continuous?.- 5.2 Derivatives and the Intermediate Value Property.- 5.3 The Mean Value Theorem.- 5.4 A Continuous Nowhere-Differentiable Function.- 5.5 Epilogue.- 6 Sequences and Series of Functions.- 6.1 Discussion: Branching Processes.- 6.2 Uniform Convergence of a Sequence of Functions.- 6.3 Uniform Convergence and Differentiation.- 6.4 Series of Functions.- 6.5 Power Series.- 6.6 Taylor Series.- 6.7 Epilogue.- 7 The Riemann Integral.- 7.1 Discussion: How Should Integration be Defined?.- 7.2 The Definition of the Riemann Integral.- 7.3 Integrating Functions with Discontinuities.- 7.4 Properties of the Integral.- 7.5 The Fundamental Theorem of Calculus.- 7.6 Lebesgue's Criterion for Riemann Integrability.- 7.7 Epilogue.- 8 Additional Topics.- 8.1 The Generalized Riemann Integral.- 8.2 Metric Spaces and the Baire Category Theorem.- 8.3 Fourier Series.- 8.4 A Construction of R From Q.